Minimize the cost of building an enclosure

[Jon89]

Problem: The manager of a store wants to build a 600 square foot rectangular enclosure. Three sides of the enclosure will be built of wood, at a cost of $14 per foot. The fourth side will be built of cement blocks, at a cost of $28 per foot. Find the dimensions of the enclosure that will minimize the cost.

What I have: I set the sides equal to L and W. I changed W to equal 600/L. Aside from that, I don't know where to go. I have done problems where a price was set, but this does not have a price limit, just find the minimum. I would greatly appreciate any and all help. Thank You!

 

[daon]

1) A=L*W=600
2) C=14*Wood + 28*Cement

Note also that since 3 sides are wood, we will say that L,L,W are all made of wood, and the last W is made of cement. That is, 2L+W will be made of wood and the remaining W to be made out of cement. Realize that our choice of W to be our 'odd-side-out' is completely arbitrary. We can just as easily do the opposite.

So, given (2),
C = 14(2L+W) + 28W

And, by (1), L = 600/W, so:
C = 14*(1200/W + W) + 28W, or:
C = 16800/W + 42W

Take derivative of Cost with respect to the "footage" of wood and set equal to zero:
dC/dW = -16800/W^2 + 42 = 0
W^2 = 400, W=20

So, L = 600/W, therefore L = 30

And our final cost is C = 14*(2L+W) + 28*W = 14*(2*30+20) + 28*20 = 1680.

-Daon